# Proving that non-regular languages are closed under concatenation

How can I prove that non-regular languages are closed under concatenation using only the non-regularity of $L=\{a^nb^n|n\ge1\}$ ?

Let $X\subseteq \mathbb{N}$ be any undecidable set containing $1$ and every even number. For example, take your favourite undecidable set $S$ and let $$X = \{0, 2, 4, \dots\} \cup \{1\} \cup \{2i+1\mid i\in S\}\,.$$ The language $\mathcal{L} = \{a^i\mid i\in X\}$ is undecidable, so it certainly isn't regular. But $$\mathcal{L}\cdot\mathcal{L} = \{a^{i+j}\mid i,j\in X\} = \{a^i\mid i\in\mathbb{N}\}\,,$$ is regular.
Here is another example showing that the claim is false. Let $L$ be any non-regular language over $\{0,1\}$, and take $$L_0 = \{ 0 w : w \in L \} \cup \{ 1 w : w \in \{0,1\}^* \} \cup \{\epsilon\}, \\ L_1 = \{ 1 w : w \in L \} \cup \{ 0 w : w \in \{0,1\}^* \} \cup \{\epsilon\}.$$ You can check that $L_0,L_1$ are not regular, but $L_0L_1 = \{0,1\}^*$ is regular.
Another nice example uses Lagrange's four square theorem, which states that every non-negative integer is a sum of four squares. Define $L_1 = \{1^{n^2} : n \geq 0\}$, $L_2 = L_1^2$ and $L_4 = L_2^2$. The four square theorem shows that $L_4 = 1^*$ is regular. Conversely, the pumping lemma shows that $L_1$ is not regular. Hence either $L_2$ is regular, in which case $L_4 = L_2^2$ is a counterexample to the claim, or $L_2$ is not regular, in which case $L_2 = L_1^2$ is a counterexample to the claim. (In fact, $L_2$ is not regular.)
Yet another example assumes the Goldbach conjecture: every even integer larger than 4 is the sum of two odd primes. Let $L = \{ 1^p : \text{$p$is an odd prime } \}$. Using the pumping lemma it is not hard to show that $L$ isn't regular. Assuming the Goldbach conjecture, $L^2 = \{ (11)^n : n \geq 3 \}$, which is regular.